Press forming is a processing method in which a die is pressed onto a blank (metal sheet) to be formed, whereby the shape of the die is transferred to the blank. In this press forming, what is called springback may occur in which the blank having been deformed slightly recovers after a press-formed product is removed from the die, which may make the shape of the press-formed product different from the desired shape. Accordingly, in the press forming, it is necessary to predict the amount of springback of the press-formed product and to design the shaped of the die so that the shape of the press-formed product after the springback becomes the desired shape on the basis of the prediction result.
The springback due to removal of stress applied in the processing occurs when the press-formed product is removed from the die. The springback will be described in more detail with reference to FIG. 16. FIG. 16 is a diagram illustrating a relation between strain and stress applied onto a material in a press-forming process and in a springback process, with strain on the horizontal axis and stress on the vertical axis. As depicted in FIG. 16, when an external force a is applied onto the material in the press-forming process, after the material undergoes the elastic deformation region, plastic deformation occurs starting at the yield point A, and the plastic deformation continues to the point B where the strain amount ε2 (stress σ2) corresponds to the desired shape. When the material is removed from the die, the external force is unloaded and the stress a decreases. This unloading ends at the point C where forces acting on the entire material are balanced with the strain amount ε1 (stress σ1).
The amount of springback is determined by the difference in the strain amount ε generated in this unloading process, i.e., the difference Δε between the strain amount ε2 at the unloading-start point B and the strain amount ε1 at the unloading-end point C. In a classic mathematical model called a conventional isotropic-hardening model, because it is assumed that the region from the unloading-start point B to the point D where the absolute value of the stress σ2 is equal to that of the unloading start point B is in an elastic deformation region, in other words, a region in which the relation between stress and strain is linear, the unloading-end point should be the point E. However, in many actual materials, such a linear region hardly exist in the unloading process and a yield phenomenon occurs much earlier than the point D, deviating from the elastic deformation region, so that the relation between stress and strain (stress-strain relation) represents a non-linear curve.
Such an early yield phenomenon after the stress reversal is called Bauschinger effect. To reproduce the Bauschinger effect, kinematic hardening needs to be considered. The kinematic hardening means hardening with a yield surface moving without changing its area. Representative examples considering the kinematic hardening include Yoshida-Uemori model (see Non Patent Literature 1). The Yoshida-Uemori model can reproduce the Bauschinger effect. Furthermore, in the Yoshida-Uemori model, the non-linear stress-strain relation immediately after the stress reversal is linearly approximated as an apparent gradient of stress versus strain (apparent Young's modulus) on the assumption that work hardening occurs linearly.
However, the behavior of the non-linear stress-strain relation in the unloading process is apparently different from the behavior obtained by linearly approximating this relation, and thus the stress-strain relation cannot be accurately reproduced by the Yoshida-Uemori model. In view of such a background, Patent Literature 1 describes a method for expressing the Bauschinger effect that occurs in an initial stage of the unloading process. In this method, the stress at the beginning of plastic deformation in the unloading process is identified based on the stress-strain gradient to make the stress at the yield point A (yield stress) lower than that in a conventional technique. Specifically, in this technique, the Bauschinger effect that occurs in an initial stage of the unloading process is expressed by reducing the elastic region in which the stress-strain relation is linear and increasing the non-linear work-hardening region.
In the method described in Patent Literature 1, to improve the accuracy in the work-hardening (plastic-deformation) region after yielding again during the unloading, a coefficient (or parameter) of saturation speed of kinematic hardening of a yield surface is defined as a function of equivalent plastic strain. This method assumes that, in the stress-strain gradient, the saturation speed is high when the stress rapidly increases in a region where the strain is small, and the saturation speed is low when the strain is high and the stress does not increase so much.